Problem 13 - Intermediate Algebra

Question

For Problems \(1-34\), solve each equation. $$ 4^{x}=8 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = \frac{3}{2}\).

Step 1: Express Both Sides with the Same Base

To solve the equation \(4^x = 8\), start by expressing both sides of the equation with the same base. Notice that 4 is \(2^2\) and 8 is \(2^3\). So, rewrite the equation as \((2^2)^x = 2^3\).

Step 2: Simplify the Exponents

Apply the power of a power rule in exponents: \((a^m)^n = a^{m \cdot n}\). This transforms the left side of the equation into \(2^{2x}\). The equation now becomes \(2^{2x} = 2^3\).

Step 3: Set the Exponents Equal

Since the bases are the same, set the exponents equal to each other: \(2x = 3\).

Step 4: Solve for x

Divide both sides of the equation by 2 to solve for \(x\): \(x = \frac{3}{2}\).

Key Concepts

Understanding Exponent Properties

Exponents are a way to represent numbers multiplied by themselves. When talking about exponent properties, we are referring to the rules that govern how exponents can be manipulated. It's important to know these properties as they make complex calculations much easier.

One key property is the rule for multiplying powers with the same base, which states that you can add exponents if the bases are the same: \(a^m \cdot a^n = a^{m+n}\). This helps simplify expressions quickly.

There is also a property that allows division of powers with the same base. When dividing powers with identical bases, you subtract the exponents: \(\frac{a^m}{a^n} = a^{m-n}\). This turns cumbersome fractions into simpler expressions.

Recognizing these properties facilitates solving larger problems step-by-step by breaking them down into more manageable pieces.

The Power of a Power Rule

The power of a power rule is another important property of exponents. This rule states that when you raise an exponent to another exponent, you multiply the exponents together: \((a^m)^n = a^{m \cdot n}\).

This is particularly useful in equations where you have nested exponents. For example, consider \((2^2)^x\); by applying the power of a power rule, it becomes \(2^{2x}\).

This simplification works because exponentiation is a form of repeated multiplication. The power of a power rule helps us streamline expressions, especially when trying to solve equations by creating equivalent expressions with the same base.

Steps for Solving Exponential Equations

Solving exponential equations involves a few strategic steps that use the properties of exponents to simplify the task. Let's break it down:

• Finding a Common Base: Try to express both sides of the equation using the same base. This often involves rewriting numbers as powers of a smaller number. For example, rewrite \(4 = 2^2\) and \(8 = 2^3\).
• Simplifying the Equation: Apply the power of a power rule where necessary to consolidate exponents. This turns expressions like \((2^2)^x\) into \(2^{2x}\).
• Setting Exponents Equal: Once you have the same base on both sides, you can set the exponents equal to each other. This step significantly simplifies the equation.
• Solving for the Variable: Finally, solve the resulting simple equation for the unknown variable.

By following these steps carefully, you can solve exponential equations efficiently